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Chapter 8: Problem 69

A Gallup poll asked a random samples of Americans in 2016 and 2018 if they were satisfied with the quality of the environment. In 2016 , 543 were satisfied with the quality of the environment and 440 were dissatisfied. In 2018,461 were satisfied and 532 were dissatisfied. Determine whether the proportion of Americans who are satisfied with the quality of the environment has declined. Use a \(0.05\) significance level.

Short Answer

Expert verified
Once the P-value is calculated and compared to the significance level, a conclusion can be drawn about whether the proportion of Americans who are satisfied with the environment quality has decreased from 2016 to 2018.

Step by step solution

01

Set Up Hypotheses

First, establish your null and alternative hypotheses. The null hypothesis assumes that there was no change in the proportion of Americans who are satisfied with the environment quality between the two years \(H_0: P_{2016} = P_{2018}\). The alternative hypothesis states that the proportion of Americans satisfied with the environment quality decreased in 2018 from 2016 \(H_1: P_{2016} > P_{2018}\).
02

Calculate Sample Proportions

Next, calculate the sample proportions for both years. For 2016, the proportion is \(P_{2016} = \frac{543}{(543+440)}\) and for 2018, the proportion is \(P_{2018} = \frac{461}{(461+532)}\).
03

Calculate Test Statistic

Calculate the test statistic using the following formula: \(Z = \frac{(P_{2016}-P_{2018}) - 0}{\sqrt{P(1-P)(\frac{1}{n_{2016}} + \frac{1}{n_{2018}})}}\) where \(P = \frac{n_{2016}P_{2016} + n_{2018}P_{2018}}{n_{2016} + n_{2018}}\). Calculate the combined proportion \(P\), then substitute all calculated values into the formula and solve for \(Z\).
04

Find P-value

Using z-table or calculator, find the P-value associated with the calculated Z score. This P-value represents the probability to observe such extreme proportions, assuming the null hypothesis is true.
05

Make Conclusion

Compare the P-value to the significance level. If the P-value is less than the significance level, reject the null hypothesis in favor of the alternative. If the P-value is greater than the significance level, do not reject the null hypothesis

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Most popular questions from this chapter

A true/false test has 50 questions. Suppose a passing grade is 35 or more correct answers. Test the claim that a student knows more than half of the answers and is not just guessing. Assume the student gets 35 answers correct out of \(50 .\) Use a significance level of \(0.05 .\) Steps 1 and 2 of a hypothesis test procedure are given. Show steps 3 and 4, and be sure to write a clear conclusion. Step $$\text { 1: } \begin{aligned}&\mathrm{H}_{0}: p=0.50 \\\&\mathrm{H}_{\mathrm{a}}: p>0.50\end{aligned}$$ Step 2: Choose the one-proportion \(z\) -test. Sample size is large enough, because \(n p_{0}\) is \(50(0.5)=25\) and \(n\left(1-p_{0}\right)=50(0.50)=25\), and both are more than \(10 .\) Assume the sample is random and \(\alpha=0.05\).

A multiple-choice test has 50 questions with four possible options for each question. For each question, only one of the four options is correct. A passing grade is 35 or more correct answers. a. What is the probability that a person will guess correctly on one multiple- choice question? b. Test the hypothesis that a person who got 35 right out of 50 is not just guessing, using an alpha of \(0.05\). Steps 1 and 2 of the hypothesis testing procedure are given. Finish the question by doing steps 3 and 4 . Step 1: \(\quad \mathrm{H}_{0}: p=0.25\) \(\mathrm{H}_{\mathrm{a}}: p>0.25\) Step 2: Choose the one-proportion \(z\) -test. \(n\) times \(p\) is 50 times \(0.25\), which is \(12.5\). This is more than 10 , and 50 times \(0.75\) is also more than 10 . Assume a random sample.

The researchers in a Pew study interviewed two random samples, one in 2015 and one in 2018 . Both samples were asked, "Have you read a print book in the last year?" The results are shown in the table below. $$\begin{array}{|lrrl|}\hline \text { Read a print book } & \mathbf{2 0 1 5} & \mathbf{2 0 1 8} & \text { Total } \\ \hline \text { Yes } & 1201 & 1341 & 2542 \\\\\hline \text { No } & 705 & 661 & 1366 \\ \hline \text { Total } & 1906 & 2002 & \\\\\hline\end{array}$$ a. Find and compare the sample proportions that had read a print book for these two groups. b. Find a pooled estimate of the sample proportion. c. Has the proportion who read print books increased? Find the observed value of the test statistic to test the hypotheses \(\mathrm{H}_{0}: p_{2015}=p_{2018}\) and \(\mathrm{H}_{\mathrm{a}}: p_{2015}

A psychologist is interested in testing whether offering students a financial incentive improves their video-game-playing skills. She collects data and performs a hypothesis test to test whether the probability of getting to the highest level of a video game is greater with a financial incentive than without. Her null hypothesis is that the probability of getting to this level is the same with or without a financial incentive. The alternative is that this probability is greater. She gets a p-value from her hypothesis test of \(0.003 .\) Which of the following is the best interpretation of the p-value? i. The p-value is the probability that financial incentives are not effective in this context. ii. The p-value is the probability of getting exactly the result obtained, assuming that financial incentives are not effective in this context. iii. The p-value is the probability of getting a result as extreme as or more extreme than the one obtained, assuming that financial incentives are not effective in this context. iv. The p-value is the probability of getting exactly the result obtained, assuming that financial incentives are effective in this context. \(\mathrm{v}\). The p-value is the probability of getting a result as extreme as or more extreme than the one obtained, assuming that financial incentives are effective in this context.

An immunologist is testing the hypothesis that the current flu vaccine is less than \(73 \%\) effective against the flu virus. The immunologist is using a \(1 \%\) significance level and these hypotheses: \(\mathrm{H}_{\mathrm{o}}: p=0.73\) and \(\mathrm{H}_{\mathrm{a}}: p<0.73\). Explain what the \(1 \%\) significance level means in context.
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